The internet has been talking about this red vs blue button problem. Since I’ve been talking about game theory lately, why not talk about this one too? You know, as a treat?

Here’s the problem. Everyone in the world is presented with a choice between a red button and a blue button. If the majority of people press the blue button, then everyone lives. If the majority of people press the red button, then only people who pressed red live, while all the people who pressed the blue button die. Which do you press?

As always, I find it funny that these questions posit life or death stakes. What if instead of dying, people were just mildly inconvenienced? Like, if they were forced to do nothing for five minutes, would that change how we approach the problem? I guess if it were framed that way, then it would be obvious that it’s not worth arguing about for more than five minutes. But anyways…

As a prisoner’s dilemma

I’m going to compare to a few other games commonly discussed in game theory. The first is the Prisoner’s Dilemma. The red button is defecting in a prisoner’s dilemma, because it leads to a selfishly beneficial outcome (you guarantee your own survival). But when most people defect, it leads to a worse overall outcome (some people die).

But it’s not purely a prisoner’s dilemma.  If we’re above 50% red, then picking red doesn’t just lead to the most selfishly beneficial outcome (you survive), it also leads to the best overall outcome (one fewer death).  And if everyone picks red, that’s just as good as everyone picking blue. The outcome follows this curve:

X axis: Percentage of people that pick the red button. Y axis: Percentage of people that die.

But if you squint at the curve, then yes the red button does have overall worse outcomes. We can formalize “squinting” by introducing some noise into the outcome. (In game theory terminology, we introduce a “trembling hand”.) Noise could come from many places. For instance, there might be some people who just randomly push buttons. The noise could also be epistemological in nature–however many people you think will pick red, your prediction will be somewhat off from reality. The effect of noise is to smear the curve, like so:

X axis: Percentage of people that pick the red button before adding noise. The noise is a random error in the final percentage that pick red. Y axis: Percentage of people who die, averaged over the probability distribution of the noise.

Once we have a smeared curve, it becomes clear that trying to make everyone pick red is overall worse than trying to make everyone pick blue. It’s because you just can’t control or predict what everyone will do.

As a coordination game

Suppose we play a game where we each pick a side of the road–either the left, or right. If we both pick the same side, then we safely drive past each other, hooray. If we pick opposite sides then we crash. Which side do you choose? It’s not much of a deep ethical question, because obviously the answer is to choose whatever everyone else is choosing.

So when I look at the outcome curve for the red/blue button, the most important aspect of the curve is that bump in the middle.  That bump represents a lot of deaths.  More than anything, you want to avoid that bump, which means getting everyone on the same side. So, just like when you choose which side of the road to drive, the correct answer is to choose the same thing as everyone else.

That’s right. Hot take! People who say red button are wrong. People who say blue button are wrong. The answer is neither. Just pick whatever everyone else is picking. Unless you have influence over a lot of other people, you should just take other people’s choices as a immutable fact, and work around it.

On the other hand, if we have time to prepare and communicate with each other, here’s a different strategy. Create a single authoritative poll, and tell everyone to commit to whatever the poll says most people will do. And then in the poll, vote blue.

The presence or absence of communication can make a difference in coordination games. Consider the following problem. There are a hundred quarters on a table, and one penny. We each silently pick a coin, and if we both pick the same coin then we win that coin. Without communication, we have little chance of landing on the same quarter. So the correct strategy is to pick the penny, because that’s easier to coordinate even though the payoff is much smaller than a quarter. On the other hand, if we could communicate, then the correct strategy is to arbitrarily agree on one of the quarters.

So, an open question. Could it be the case that if we can communicate, the correct choice is blue, but if we can’t communicate, then the correct choice is red?

As a battle of the sexes

The battle of the sexes is a special kind of coordination game. In the battle of the sexes, a married couple can’t decide where to go out for the night. One of them wants to go to the wrestling match. His husband wants to go to the opera. Both of them want their own way, but above all they prefer to go together.

A good strategy in the battle of the sexes is to somehow convince the other player that you are committed to your own choice, much more committed than they are to their own choice. For example, buy some non-refundable tickets to the wrestling match. But suppose that commitment isn’t possible–all tickets are fully refundable.

So the strategy is to act like you are irrationally committed to your choice. Declare that you will go to the wrestling match with or without your husband, even though in reality you would hate to go alone. But you could just be pretending to be irrational, and your husband knows it is in your interest to pretend. So you got to pretend even harder, and your husband has to ignore your pretense even harder. On and on in an ever-increasing arms race. Basically it’s an unhealthy marriage, and you should just divorce your game theorist husband already.

The red vs blue button problem is basically a battle of the sexes. Some people have a philosophical preference for red, and some have a philosophical preference for blue. Each group prefers their own answer, but above all they need to reach a consensus. What’s the correct strategy? I suppose you should make a big deal about how irrationally committed you are to your own answer. And the rest of us should ignore you because we all know it’s in your interest to pretend.  Then you should pretend harder, and we should ignore you harder, in an ever-increasing arms race.

In other words, game theory says the correct strategy is to argue loudly and meaninglessly on the internet. I think most people don’t know that much about game theory, but in this one case they’re nailing it!